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Wall-crossing and holomorphicanomaly

Hamburg, December 2, 2010

Albrecht Klemm

M. Alim , B. Hagighat, M. Hecht, A.K, M. Rauch, T. Wotschke Dec 2010

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∙ Objective of the work

– Generalize holomorphic anomaly from string- to gauge

theory,...

– Topological String as an example

∙ M5 brane(s) on a divisor in a CY 3-fold

– Elliptic genus of the MSW string

– N = 4 SYM theory perspective

∙ Wall-crossing for D4-D2-D0 branes

– The Kontsevich-Soibelmann formula

– Gottsche’s Wall-crossing formula

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∙ The holomorphic anomaly equation

– Zwerger’s anholomorphic regularisation of indefinite Θ

-fcts

– Proof of the holomophic anomaly for r = 2

∙ Conclusions

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Objective of the work:

∙ Generalize the holomorphic anomaly equation capturing

background dependence of topological string theory.

∙ Extend the relation between modularity and

holomorphicity → Mock modular objects .

∙ Interprete the failure of holomorphicity in terms of Wall

crossing.

∙ Proof a holomorphic anomaly equation for the partition

function of N = 4 SYM on del Pezzo surfaces.

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Topological String on CY M as example:

Key object of interest in any topological theory:

Topological partition function

Z(t, t) = exp

⎛⎝ ∞∑g=0

�2g−2Fg(t, t)

⎞⎠

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∙ Fg(t, t) invariant under space-time duality group Γ ∈SP (ℎ3(M),ℤ)

∙ Fg(t, t) yields generating function for M2/D2+D0 BPS

invariants n�g ∈ ℤ in holomorphic limit∑g �

2g−2Fg(t) =∑g

�2g−2 limt→t∞

Fg(t, t)

=∑

m,�∈H2(Mℤ)

n�gm

(2 sin

(k�

2

))2g−2

emt⋅�

t∞ reflects the background dependence of the BPS

numbers n�g ∈ ℤ

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∙ Fg(t, t) fullfill an holomorphic anomaly equation

∂tkFg =

∫ℳ(g)

∂∂�g =1

2Cij

k

⎛⎝DiDjFg−1 +

g−1∑r=1

DiFrDjFg−r

⎞⎠ .

a b

dc

=

c

=

a

d

b

c

j

φi

jφ

Σ ij η

ba

dΣij

φ φ ji η= φ

φ j

i

Σij

(−1)F

= φφ φi jΣij

η

ηij

ijij

ij

∙ The anholomorphicity comes from the boundary of the

moduli space ℳ(g)

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Solution

∙ Fg(t, t) are generated by finite polynomials rings of

quasimodular forms Γ ∈ SP (ℎ3(M),ℤ) of weight 3g−3

→ finite numbers of unknows.

∙ Anholomorphic generators are analogs of the

anholomorphic second Eisenstein Series E2(�, �) =

E2(�)− 3�Im(�).

∙ Boundary conditions for limt→tcrit fix the unknows.

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M5 branes on a divisor

Wrap M5-brane(s) r-times on a divisor P in a Calabi-Yau

3-fold M and extended in S1 × ℝ3,1

Two effective descriptions of the effective action of the

M5-brane

∙ Dimensional reduction of the effective action on M5

on small P leads to the MSW string with (0,4) CFT

worldsheet theory. �-model whose target space are the

def. of M5 on P . Maldacena, Strominger, Witten 97 r = 1.

∙ Reduction of the effective action on M5 on P × T2

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with small T2 leads to N = 4 U(r) SYM on P .

SL(2,ℤ) invariance of SYM is geometrized on the

complex modulus of the torus � = 4�ig2

+ �2�. Minahan,

Nemeschansky, Warner, Vafa 98

BPS states of the MSW string can be computed by

compatifying M5 on P × T2 as elliptic genus of the

MSW string partition function obeying SL(2,ℤ)

invariance and compared with the topological partition

function of N = 4 U(r) and after splitting of U(1)

SU(r) gauge theory.

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Elliptic genus of the MSW string

Degrees of freedom (scalars):

∙ R/L scalars �A from reduction of selfdual 3-form ℎ(3) =

d�AR/L ∧ �±A, �±A ∈ H2

±(P,ℤ),

∙ ℎ0(M,ℒP) − 1 right scalars, describing movement of

M5 in M. In this work we consider rigid divisors. It

implies P a del Pezzo surface (b+2 = 1).

∙ right scalars in the center of mass multiplett, describing

movement in ℝ3,1.

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Symmetries and charges (type IIA picture):

∙ rpA D4 brane charge

∙ M2 brane gives rise to

– qA: D2 brane charges in Λ = H2±(X,ℤ)∣P .

– q0 momentum of the M2 around the S1 ∈ T2

The charges are

Γ = (Q6, Q4, Q2, Q0) = r(0, pA, qA, q0),

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One defines the modified elliptic genus

Z ′(r)X,P(�, z) = TrℋRR

(−1)FR F 2R q

L′0−cL24 qL

′0−

cR24e2�iz⋅Q2,

which is expected (proven for r = 1) to be a (0, 2)

((−32,

12) after seperating the CMM) SL(2,ℤ) Jacobiform,

with modular parameter � and elliptic parameter z.

It receives contributions only from states annihilated by(L0 −

cR

24− 1

2q2R

)∣q⟩ = 0 .

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Using the spectral flow symmetry with k ∈ Λ

− q0 7→ −q0 + k ⋅ q +1

2k ⋅ k,

q 7→ q + k,

one can show using the symmetries of d(r,Q,Q0)

Z(r)X,P(�, z) =

∑Q0;QA

d(r,Q,Q0) e−2�i�Q0 e2�iz⋅Q2

=∑

q0;q∈Λ∗+[P ]2

d(r, q,−q0) e−2�i�rq0 e2�irz⋅q,

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for arbitrary rank the decomposition with q0 = −q0 − 12q

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Z(r)X,P(�, z) =

∑�∈Λ∗/Λ

f (r)� (�)�(r)

� (�, z), (1)

f (r)� (�) =

∑q0≥−

cL24

d(r)� (q0)e

2�i�rq0,

�(r)� (�, z) =

∑k∈Λ+

[P ]2

(−1)rp⋅(k+�)e2�i�r(k+�)2

2 e2�irz⋅(k+�).

Note the shift [P ]2 due to the Freed-Witten anomaly,

whenever P is not a spin manifold.

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N = 4 gauge theory perspective

The generating function

f(r)�,J(�) =

∑d≥ 0

(−1)rp⋅� Ω(Γ; J) qd−r�(P )24

of the BPS invariants Ω(Γ; J), defined through

Ω(Γ; J) =∑m∣Γ

Ω(Γ/m; J)

m2

in terms of Euler# of moduli sp. of coherent sheaf on P

Ω(Γ, J) = (−1)dimℂℳJ(Γ)�(ℳ(Γ), J)

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Coherent sheaves represent the gauge theory instantons

configurations of N = 4 SYM theory Vafa, Witten 94.

They depend in general through Wall-Crossing behaviour

on the Kahlerclass J in the Kahlercone C(P ) on P .

Also the Θ-functions depend on the Kahlerparameter J

through a choice of polarization:

k2+ =

(k ⋅ J)2

J ⋅ J, k2

− = k2 − k2+.

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�(r)�,J(�, z) =

∑k∈Λ+

[P ]2

(−1)rp⋅(k+�)e2�i�r(k+�)2+

2 e2�i�r(k+�)2−

2 e2�irz⋅(k+�),

rank=1: L. Gottsche 97∑q0

d(1, �, q0) e2�i� q0 =

1

��(P ).

For a single M5-brane Ω and Ω become identical and

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independent of J . This is reflected by the holomorphicity(∂� +

1

4�i∂2z+

)Z

(1)X,P(�, z) = 0.

For multiple M5-branes and if the branes cannot seperate

(rigid case) one expects boundstates at threshold and a

holomorphic anomaly on the rhs. Minahan, Nemeschansky, Warner,

Vafa 98. A famous example for the not rigid case is P is a

K3. Here we get the elliptic genus of the heterotic string

which is holomophic.

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Wall-crossing for D4-D2-D0 branes

In the large volume limit the even D-branme charges are

given by the topological data of the sheaf ℰ on P

Γ = r

(0, [P ], i∗F (ℰ),

�(P )

24+

∫P

1

2F (ℰ)2 −Δ(ℰ)

),

with

Δ(ℰ) :=1

r(ℰ)

(c2(ℰ)− r(ℰ)− 1

2r(ℰ)c1(ℰ)2

), �(ℰ) :=

c1(ℰ)

r(ℰ), F (ℰ) := �(ℰ)+

[P ]

2.

Given a choice of J , a sheaf ℰ is called �-semi-stable if

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for every sub-sheaf ℰ ′

�(ℰ ′) ⋅ J ≤ �(ℰ) ⋅ J

Walls of marginal stability in J are given by

�(ℰ ′) ⋅ J = �(ℰ) ⋅ J

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The Kontsevich-Soibelmann wall-crossing formula

Define a symplectic pairing

⟨Γ1,Γ2⟩ = r1r2(�2 − �1) ⋅ [P ] ,

a Lie algebra

[eΓ1, eΓ2] = (−1)⟨Γ1,Γ2⟩⟨Γ1,Γ2⟩eΓ1+Γ2.

and Lie group elements

UΓ = exp

⎛⎝−∑n≥1

enΓ

n2

⎞⎠ . (2)

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The Kontsevich-Soibelman wall-crossing formula

↷∏Γ:Z(Γ;J)∈V

UΩ(Γ;J+)Γ =

↷∏Γ:Z(Γ;J)∈V

UΩ(Γ;J−)Γ , (3)

where J+ and J− denote Kahler classes on the two sides

of the wall, V is a region in IR2 bounded by two rays

starting at the origin and ↷ denotes a clockwise ordering

of the central charges.

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Rank 2: All commutators with rank > 2 vanish and with

the Baker-Campell-Hausdorff one gets the primitive

wall-crossing formula

ΔΩ(Γ) = (−1)⟨Γ1,Γ2⟩−1⟨Γ1,Γ2⟩∑

Q0,1+Q0,2=Q0

Ω(Γ1) Ω(Γ2).

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Gottsche’s wall crossing formula:

Idea: Count invariants of sheaves on surfaces with

b+2 = 1 using the wall-crossing behaviour:

∙ Provide counting function in one chamber

– by a vanishing theorem

– using blow-up formulas

∙ Use the primitive wall-crossing formula

∙ Sum over (infinitly many) walls to obtain indefinite

Theta function

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Program completed for rank 2. Let Γ = (2, �, d),

d = d1 + d2 + � ⋅ � where � = �1 − �. A wall is given by

W � = {J ∈ C(P ) ∣ � ⋅ J = 0}. From primitive

wall-crossing formula:

∑d≥ 0

(Ω(Γ; J+)− Ω(Γ; J−))qd−�(P )/12

=∑

d1,d2≥ 0, �

(−1)2�⋅[P ] � ⋅ [P ] Ω(Γ1)Ω(Γ2)qd1+d2+�2−2�(P )

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= (−1)2�⋅[P ]−1 1

�(�)2�(P )

∑�

� ⋅ [P ] q�2,

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= (−1)2�⋅[P ]−1 1

�(�)2�(P )Coeff2�iy(Θ

J+,J−Λ,� (2�, [P ]y))

Here

ΘJ,J ′

Λ,� (�, x) :=1

2

∑�∈Λ+�

(sgn(J ⋅ �)− sgn(J ′ ⋅ �)) q12�

2e2�i�⋅x.

These theta functions obey a cocycle condition

ΘF,GΛ,� + ΘG,H

Λ,� = ΘF,HΛ,� ,

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Let J and J ′ be in arbitrary chambers∑d≥0(Ω(Γ; J)−Ω(Γ; J ′))qd−�(P )/12

=(−1)2�⋅[P ]

�(�)2�(P )Coeff2�iy(Θ

J,J ′

Λ,� (2�, [P ]y)).

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The holomorphic anomaly equation

Idea: Regularisation of the indefinite Θ-function

ΘJ,J ′

Λ,� (�, x) :=1

2

∑�∈Λ+�

(sgn(J ⋅ �)− sgn(J ′ ⋅ �)) q12�

2e2�i�⋅x.

is incompatible with the SL(2,ℤ) invariance of N = 4

SYM.

One has to use a modular regularisation of the theta

function. Such a regularisation has been provided by

Zwegers treatment of Mock-modular forms Zwegers 09. It is

non-holomorphic an was used to construct an

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anholomophic extension to make Ramanujan’s Mock

modular forms into modular forms.

This idea appeared in similar context in the work of Jan

Manschot Manschot 08 & 09

Using the an-holmophicity of Zwergers regularistion we

will prove the holomorphic anomaly equation.

Zwerger’s anholomorphic regularisation of indefinite

Θ-fcts

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f(2)�,J ′(�)− f (2)

�,J(�) =#Λ⊥(�)2

�2�(P )(�)Coeff2�iy(Θ

J,J ′

Λ,� (2�, [P ]y)).

transforms not well under SL(2,ℤ).

Zwegers regularization:

Θc,c′

Λ,�(�, x) = 12

∑� ∈Λ+�

((E

((c⋅(�+Im (x)

�2))√�2√

−Q(c)

)−

E

((c′⋅(�+Im (x)

�2))√�2√

−Q(c′)

))e2�i�⋅xqQ(�),

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here Q(x) = 12x

2 and E denotes the incomplete error

function

E(x) = 2

∫ x

0

e−�udu.

The non-holomorphic function Θc,c′

Λ,�(�, x) transforms as a

Jacobi form of weight 12r(Λ).

Using

E(x) = sgn(x)(1− �12(x2)),

one splits

Θc,c′

Λ,�(�, x) = Θc,c′

Λ,�(�, x)− Φc�(�, x) + Φc′

�(�, x),

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with

Φc�(�, x) =

1

2

∑� ∈Λ+�

⎡⎣sgn(� ⋅ c)− E

⎛⎝(c ⋅ (� + Im (x)�2

))√�2√

−Q(c)

⎞⎠⎤⎦ e2�i�⋅xqQ(�).

containing all anholomorphicity. Using ideas review in

Zagier 07 one shows

∂�Coeff2�iyΦc�(2�, [P ]y) = −�

−32

2

8�i

c ⋅ [P ]√−c2

(−1)4�2 �(2)�,c(�, 0),

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Splitting

f(2)�,J(�) = f

(2)�,J ′(�)− 1

�2�(P )Coeff2�iyΘ

J,J ′� (2�, [P ]y),

into the holomophic ambiguity f(2)�,J ′(�) and writting the

reduced elliptic genus (spitting the center of mass mode

off)

Z(2)X,P(�, z) =

∑�∈Λ∗/Λ

f(2)�,J(�)�

(2)�,J(�, z).

With

Dk = ∂� +i

4�k∂2z+,

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one proofs the holomorphic anomaly equation at rank

two for general surfaces P

D2Z(2)X,P(�, z) = �

−3/22

1

16�i

J ⋅ [P ]√−J2

(Z

(1)X,P(�, z)

)2

.

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Conclusions

∙ We generalized the holomorphic anomaly in topological

string theory counting D2-D0 brane BPS invariants to

D4-D2-D0 brane invariants associated to N = 4 gauge

theory instantons..

∙ The form that we proof is compatible with the

conjectured form for the E-string related to gauge

theory on half K3 Minahan, Nemeschansky, Warner, Vafa 98

∂�Z(n)(q1) =

i�−22

16�

n−1∑s=1

s(n− s)Z(s)(q1)Z(n−s)(q1) ,

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∙ Asuming this one can make a prediction for the general

anomaly of the f

∂f�(r) =∑n=1,�

�(n), �(r−n)

n(r − n)f�(n)f�(r−n)e

(r n (r − n)�

2�2

),

with � ∈ Λ + �(r) − �(n) + �(r−n) + P2 .

∙ Quasimodularity of the topological string gets replaced

by Mock modularity. To every mock modular form ℎ of

weight k there exists a shadow g ∈M2−k such that

ℎ(�) = ℎ(�) + g∗(�)

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transforms as a wheight k modular from. Let gc(z) =

g(−z), and g∗(�) is defined by

g∗(�) = −(2i)k∫ ∞−�

(z + �)−kgc(z) dz.

Then∂ℎ

∂�=∂g∗

∂�= �−k2 g(�).

E2 indead is a trivial case of a mock modular form with

constant shadow:

E2(�) = E2(�)− 3

��2.

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From ∂�E2 = �−22

3i2� we get g = 3i

2�

g∗(�) = − (2i)2 ∫∞−�(z + �)−2 3i

2�dz

= −6i�

[ −1z+�

]∞−� = − 3

��2.